https://r2.erweima.ai/mathbot/image/1727009682097ael3y4wh.jpg

The problem asks to determine the time it will take for all the water to drain from a hemispherical tank with a small circular hole at the bottom. Here's how we approach this problem.

### Given Data:
- **Shape**: Hemispherical tank
- **Radius of the hemisphere**: \( r = 5.3 \) feet
- **Hole diameter**: 1 inch (which is \( \frac{1}{12} \) feet)
- **Hole radius**: \( \frac{1}{24} \) feet
- The tank is initially full at \( t = 0 \).

The key formula we can use is Torricelli's Law, which relates the velocity of water draining from an orifice to the height of the water above the orifice. The law is given by:
\[
v = \sqrt{2gh}
\]
where:
- \( v \) is the speed of the water exiting the hole.
- \( g \) is the acceleration due to gravity (32.2 ft/s²).
- \( h \) is the height of the water column above the hole.

The rate at which the water exits through the hole is:
\[
\frac{dV}{dt} = -A_{\text{hole}} \cdot v
\]
where:
- \( A_{\text{hole}} \) is the area of the hole, which is \( \pi \left(\frac{1}{24}\right)^2 \) ft².
- \( v \) is the velocity of the water, as calculated using Torricelli’s Law.

### Volume of the Water
The volume of water in a hemispherical tank as a function of the water level \( y \) is:
\[
V(y) = \frac{\pi}{3} (r^2 y - \frac{y^3}{3})
\]
We can relate the change in volume over time to the flow rate.

### Integrating to Find Time
We need to integrate the equation for the change in volume to find the total time it will take for the tank to empty. I'll compute the exact steps for you based on this setup.

Let me calculate this for you.The time it will take for all the water to drain from the hemispherical tank through the small hole is approximately **469,084 seconds**, which is about **130 hours** or **5.4 days**.

Would you like more details on any part of this calculation? Here are some related questions:

1. How does Torricelli's law apply to fluid flow?
2. What are the assumptions behind this calculation?
3. How would the time change if the hole size were different?
4. How do we calculate the volume of water in a hemisphere at different levels?
5. What impact does gravity have on the draining time?

**Tip**: Numerical integration is often used in cases where direct integration isn't straightforward, especially with complex geometries like hemispheres.

Un tanque semiesférico tiene un radio superior de r = 5.3 pies (como se muestra en la figura) y en el tiempo (medido en segundos) t = 0 está lleno de agua. En ese momento se le hace un orificio circular con un diámetro de 1 pulgada en el fondo del tanque. ¿Cuánto tiempo pasará para que salga toda el agua del tanque?

Find out how to calculate the time required to empty a hemispherical tank with a 5.3 ft radius through a 1-inch diameter hole using Torricelli's Law. This fluid mechanics problem involves advanced concepts such as calculus and differential equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation